Abstract: A band of 17 pirates stole a sack of gold coins. When they tried to divide the fortune into equal portions, 3 coins remained. In the ensuing brawl over who should get the extra coins, one pirate was killed. The wealth was redistributed, but this time an equal division left 10 coins. Again an argument developed in which another pirate was killed. The fortune was now evenly distributed among the survivors. What was the least number of coins that could have been stolen and what does this ancient Chinese problem have to do with polynomials?

Abstract: In topology one studies geometric objects up to deformations. In this talk we will examine a class of examples accessible to everyone: knots. What does a mathematician mean by a knot? How can we tell if two knots are different? We will use pictures and animations to discuss these questions and more.

"Linear Chaos," Gabriel Prajitura (The College of Brockport, State University of New York)

Abstract: We will discuss some metric and topologic properties of orbits of linear operators (density, irregularity, types of hypercyclicity, self density). The behavior of an orbit is, in many ways, extreme. Usually, there is no "in - between" case. For example, an orbit is either dense or nowhere dense. We will show some other properties of the same type.

Student Colloquium: Thursday, October 2, 4:00 p.m. in Biology 101

Abstract: Algebraic Geometry, the study of the solution sets of systems of polynomial equations, arises in many application areas, such as chemical equilibrium,computer graphics, machine vision, and mechanical systems, especially robotics. It is also important in its own rights as a subject in pure mathematics and has connections to numerical analysis and partial differential equations. While systems of linear equations can have at most one irreducible solution set (an isolated point, a line, a plane, etc.),systems of polynomial equations often have multiple solutions and can even have solution components at several different dimensions. In recent years,Numerical Algebraic Geometry has emerged as a powerful tool for finding and manipulating such solution sets, thereby solving problems from engineering,science, and exploratory mathematics. Numerical Algebraic Geometry is based on polynomial continuation, wherein most of the computation involves numerically tracing out numerous algebraic curves. Since these paths can be traced out independently, the algorithms parallelize simply and efficiently, giving the approach a great advantage over algorithms based on symbolic computer algebra, which tend to parallelize poorly. This opens up the possibility of solving much larger systems than could be addressed before. This talk will outline the main algorithms of Numerical Algebraic Geometry, illustrating their use on problems arising in the kinematics of mechanisms and robotics.

A Brief Bio of the speaker: Charles Wampler is a Technical Fellow at the General Motors Research and Development Center, Warren, Michigan, and an Adjunct Professor of Mathematics at the University of Notre Dame. He has degrees in Mechanical Engineering from MIT (B.S. 1979) and Stanford University (M.S. 1980, Ph.D.1985). With longtime colleague Andrew Sommese, he is one of the prime developers of Numerical Algebraic Geometry and coauthor of the book, "The Numerical Solution of Systems of Polynomials Arising in Engineering and Science", World Scientific, 2005.

Faculty Colloquium: Tuesday, October 7, 4:00 p.m. in 372 Olin Science

Abstract: A non-trivial group G is said to be simple if its only normal subgroups are itself and the trivial subgroup. It is well-known that simple groups occupy a special place in the theory of finite groups--in a sense, all finite groups are `made from' simple groups. For infinite groups the situation is less clear cut. We'll discuss some of what is known, paying particular attention to examples old and new.

Abstract: High-speed digital communications must anticipate the possibility of channel errors in transmission. Nontrivial mathematical approaches to this date back to 1948, and constitute the subject of Algebraic Coding Theory. The principle of using the coefficients of polynomial remainders as a checksum to detect errors dates back to at least 1961; this idea blended well with the technology of shift registers, and is used in everything from the GPS system to the internet. Advances in modern technology such as Field Programmable Gate Arrays (FPGA), allow for very efficient vector addition of long binary inputs, and open new possibilities.

This talk will explain the idea of CRC checksums, show how basic abstract algebra plays a pivotal role, and finally, present a new mathematical algorithm (developed at NSA) to process them. Although the approach can be explained with some simple linear algebra, it implicitly takes advantage of an underlying quotient ring context that is usually ignored. Potential advantages for FPGA use will be discussed.

Suggested background: Linear Algebra

Algebra Seminar: Thursday, October 9, 4:00 p.m. in 372 Olin Science

Abstract: Let G be a d-generator group where d ≥ 2. In general, for each integer n ≥ d, there exist many n-element (ordered) generating sets for G. Probably the most natural way to classify these generating sets is by collecting them into the Nielsen Equivalence Classes of G on n-generators. We discuss some applications of Algebraic K1-theory to the theory of Nielsen Equivalence Classes of polycyclic groups.

Many sessions are scheduled throughout the day. The keynote speakers for each level are the following.

Elementary school -- David J. Whitin, Professor of Elementary Education and Phyllis E. Whitin, Associate Professor of Elementary Education, both from Wayne State University. They have written two books that explore how teachers and students can integrate literature and mathematics.Their talk is 9:45 - 10:55 a.m. in The Forum, 272 Langone Center.

Middle school -- Richard Kalman, Executive Director of Mathematical Olympiad for elementary and middle schools. He was a classroom teacher for 34 years and is the author of many articles on problem solving.His talk is 8:30 - 9:40 a.m. in 116 Rooke Chemistry.

High school -- John Allen Paulos, Professor of Mathematics, Temple University. He is a noted author, popular public speaker and monthly columnist of ABCNews.com.His talk is 11:15 a.m. - 12:25 p.m. in 116 Rooke Chemistry.

Abstract: While on sabbatical last year in Germany, my son posed the question whether family names eventually die out. In other words, if a reincarnation of Charlie Houdenecker is born 10,000,000 years after Charlie dies, could he meet one of his own descendants with the last name of Houdenecker? A simple back of the Bierdeckel computation showed that the answer depends on a certain assumption about the fertility of Charlie and his descendants. Surprisingly, this problem has a history and is of some interest to biologists. We will discuss the mathematics of the problem and a bit of its history.

Distinguished Visiting Professor, October 29 - November 2

Carl Cowen, Purdue University and Indiana University - Purdue University Indianapolis

Hosted by Pamela Gorkin, Karl Voss, and Ueli Daepp

Faculty Colloquium: Friday, October 31, 4:00 p.m. in 372 Olin Science

Abstract: The problem that is the focus of this talk concerns polynomials in one variable, viewed as complex valued functions on the complex plane. In addition to the polynomials being a ring, with the ring operations being addition and multiplication of polynomials in the usual way, the polynomials are also closed under composition of functions. Clearly, the composition of a polynomial of degree m with a polynomial of degree n gives a polynomial of degree mn. In this talk, we will investigate the question "When can a given polynomial be written as the composition of two non-trivial polynomials?" For example, some polynomials of degree 15 can be written as a composition of a polynomial of degree 3 and another of degree 5, and some cannot. If we are given a polynomial of degree 15, how can we tell whether it is or is not a non-trivial composite? The answer will be given in terms of a new group associated with the polynomial.

Come hear your fellow students talk about their diverse mathematical experiences both on and off campus, and get a better understanding of the summer research opportunities in Mathematics which are available to you.

Departmental Road Trip: Tuesday, November 17

On November 17, the Mathematics Department provided three busses for students, faculty, and friends to travel to State College. Everybody enjoyed Indian food and Terence Tao's talk "Long Arithmetic Progressions of Primes."

Abstract: Primary drivers for the increased use and “democratization” of statistics in industry have been the availability of user friendly software packages and the advent of “Six Sigma”, a business improvement methodology that extensively uses statistical tools and concepts. This methodology and associated tools such as regression analysis, design of experiments, optimization, and simulation techniques will be discussed and demonstrated. It is hoped that the listener will gain a sense of how statistics is used in an industrial research setting and how a career in this area of application might be developed.

Some background information: Mr. Dave Sartori is from PPG in Monroeville, PA. PPG is a leading manufacturer of coatings (e.g. paints for automobiles and aerospace, paint brands such as Olympic and Lucite) and specialty products (such as fiber glass for use on printed circuit boards, eye wear materials (e.g Transitions lenses) and specialty chemicals.

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